dist/src/root.c

/* mpfr_root, mpfr_rootn_ui, mpfr_rootn_si -- kth root.

Copyright 2005-2023 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.

This file is part of the GNU MPFR Library.

The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER.  If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */

#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"

 /* The computation of y = x^(1/k) is done as follows, except for large
    values of k, for which this would be inefficient or yield internal
    integer overflows:

    Let x = sign * m * 2^(k*e) where m is an integer

    with 2^(k*(n-1)) <= m < 2^(k*n) where n = PREC(y)

    and m = s^k + t where 0 <= t and m < (s+1)^k

    we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(k*(n-1))
    i.e. m must have at least k*(n-1)+1 bits

    then, not taking into account the sign, the result will be
    x^(1/k) = s * 2^e or (s+1) * 2^e according to the rounding mode.
 */

static int
mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k,
               mpfr_rnd_t rnd_mode);

int
mpfr_rootn_ui (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
{
  mpz_t m;
  mpfr_exp_t e, r, sh, f;
  mpfr_prec_t n, size_m, tmp;
  int inexact, negative;
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("x[%Pd]=%.*Rg k=%lu rnd=%d",
      mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
     ("y[%Pd]=%.*Rg inexact=%d",
      mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  if (MPFR_UNLIKELY (k <= 1))
    {
      if (k == 0)
        {
          /* rootn(x,0) is NaN (IEEE 754-2008). */
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else /* y = x^(1/1) = x */
        return mpfr_set (y, x, rnd_mode);
    }

  /* Singular values */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
          MPFR_RET_NAN;
        }

      if (MPFR_IS_INF (x)) /* (+Inf)^(1/k) = +Inf
                              (-Inf)^(1/k) = -Inf if k odd
                              (-Inf)^(1/k) = NaN if k even */
        {
          if (MPFR_IS_NEG (x) && (k & 1) == 0)
            {
              MPFR_SET_NAN (y);
              MPFR_RET_NAN;
            }
          MPFR_SET_INF (y);
          MPFR_SET_SAME_SIGN (y, x);
        }
      else /* x is necessarily 0: (+0)^(1/k) = +0
                                  (-0)^(1/k) = +0 if k even
                                  (-0)^(1/k) = -0 if k odd */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          MPFR_SET_ZERO (y);
          if (MPFR_IS_POS (x) || (k & 1) == 0)
            MPFR_SET_POS (y);
          else
            MPFR_SET_NEG (y);
        }
      MPFR_RET (0);
    }

  /* Returns NAN for x < 0 and k even */
  if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && (k & 1) == 0))
    {
      MPFR_SET_NAN (y);
      MPFR_RET_NAN;
    }

  /* Special case |x| = 1. Note that if x = -1, then k is odd
     (NaN results have already been filtered), so that y = -1. */
  if (mpfr_cmpabs (x, __gmpfr_one) == 0)
    return mpfr_set (y, x, rnd_mode);

  /* General case */

  /* For large k, use exp(log(x)/k). The threshold of 100 seems to be quite
     good when the precision goes to infinity. */
  if (k > 100)
    return mpfr_root_aux (y, x, k, rnd_mode);

  MPFR_SAVE_EXPO_MARK (expo);
  mpz_init (m);

  e = mpfr_get_z_2exp (m, x);                /* x = m * 2^e */
  if ((negative = MPFR_IS_NEG(x)))
    mpz_neg (m, m);
  r = e % (mpfr_exp_t) k;
  if (r < 0)
    r += k; /* now r = e (mod k) with 0 <= r < k */
  MPFR_ASSERTD (0 <= r && r < k);
  /* x = (m*2^r) * 2^(e-r) where e-r is a multiple of k */

  MPFR_MPZ_SIZEINBASE2 (size_m, m);
  /* for rounding to nearest, we want the round bit to be in the root */
  n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN);

  /* we now multiply m by 2^sh so that root(m,k) will give
     exactly n bits: we want k*(n-1)+1 <= size_m + sh <= k*n
     i.e. sh = k*f + r with f = max(floor((k*n-size_m-r)/k),0) */
  if ((mpfr_exp_t) size_m + r >= k * (mpfr_exp_t) n)
    f = 0; /* we already have too many bits */
  else
    f = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k;
  sh = k * f + r;
  mpz_mul_2exp (m, m, sh);
  e = e - sh;

  /* invariant: x = m*2^e, with e divisible by k */

  /* we reuse the variable m to store the kth root, since it is not needed
     any more: we just need to know if the root is exact */
  inexact = mpz_root (m, m, k) == 0;

  MPFR_MPZ_SIZEINBASE2 (tmp, m);
  sh = tmp - n;
  if (sh > 0) /* we have to flush to 0 the last sh bits from m */
    {
      inexact = inexact || (mpz_scan1 (m, 0) < sh);
      mpz_fdiv_q_2exp (m, m, sh);
      e += k * sh;
    }

  if (inexact)
    {
      if (negative)
        rnd_mode = MPFR_INVERT_RND (rnd_mode);
      if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA
          || (rnd_mode == MPFR_RNDN && mpz_tstbit (m, 0)))
        inexact = 1, mpz_add_ui (m, m, 1);
      else
        inexact = -1;
    }

  /* either inexact is not zero, and the conversion is exact, i.e. inexact
     is not changed; or inexact=0, and inexact is set only when
     rnd_mode=MPFR_RNDN and bit (n+1) from m is 1 */
  inexact += mpfr_set_z (y, m, MPFR_RNDN);
  MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mpfr_exp_t) k);

  if (negative)
    {
      MPFR_CHANGE_SIGN (y);
      inexact = -inexact;
    }

  mpz_clear (m);
  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inexact, rnd_mode);
}

/* Compute y <- x^(1/k) using exp(log(x)/k).
   Assume all special cases have been eliminated before.
   In the extended exponent range, overflows/underflows are not possible.
   Assume x > 0, or x < 0 and k odd.
   Also assume |x| <> 1 because log(1) = 0, which does not have an exponent
   and would yield a failure in the error bound computation. A priori, this
   constraint is quite artificial because if |x| is close enough to 1, then
   the exponent of log|x| does not need to be used (in the code, err would
   be 1 in such a domain). So this constraint |x| <> 1 could be avoided in
   the code. However, this is an exact case easy to detect, so that such a
   change would be useless. Values very close to 1 are not an issue, since
   an underflow is not possible before the MPFR_GET_EXP.
*/
static int
mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
{
  int inexact, exact_root = 0;
  mpfr_prec_t w; /* working precision */
  mpfr_t absx, t;
  MPFR_GROUP_DECL(group);
  MPFR_TMP_DECL(marker);
  MPFR_ZIV_DECL(loop);
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_TMP_INIT_ABS (absx, x);

  MPFR_TMP_MARK(marker);
  w = MPFR_PREC(y) + 10;
  /* Take some guard bits to prepare for the 'expt' lost bits below.
     If |x| < 2^k, then log|x| < k, thus taking log2(k) bits should be fine. */
  if (MPFR_GET_EXP(x) > 0)
    w += MPFR_INT_CEIL_LOG2 (MPFR_GET_EXP(x));
  MPFR_GROUP_INIT_1(group, w, t);
  MPFR_SAVE_EXPO_MARK (expo);
  MPFR_ZIV_INIT (loop, w);
  for (;;)
    {
      mpfr_exp_t expt;
      unsigned int err;

      mpfr_log (t, absx, MPFR_RNDN);
      /* t = log|x| * (1 + theta) with |theta| <= 2^(-w) */
      mpfr_div_ui (t, t, k, MPFR_RNDN);
      /* No possible underflow in mpfr_log and mpfr_div_ui. */
      expt = MPFR_GET_EXP (t);  /* assumes t <> 0 */
      /* t = log|x|/k * (1 + theta) + eps with |theta| <= 2^(-w)
         and |eps| <= 1/2 ulp(t), thus the total error is bounded
         by 1.5 * 2^(expt - w) */
      mpfr_exp (t, t, MPFR_RNDN);
      /* t = |x|^(1/k) * exp(tau) * (1 + theta1) with
         |tau| <= 1.5 * 2^(expt - w) and |theta1| <= 2^(-w).
         For |tau| <= 0.5 we have |exp(tau)-1| < 4/3*tau, thus
         for w >= expt + 2 we have:
         t = |x|^(1/k) * (1 + 2^(expt+2)*theta2) * (1 + theta1) with
         |theta1|, |theta2| <= 2^(-w).
         If expt+2 > 0, as long as w >= 1, we have:
         t = |x|^(1/k) * (1 + 2^(expt+3)*theta3) with |theta3| < 2^(-w).
         For expt+2 = 0, we have:
         t = |x|^(1/k) * (1 + 2^2*theta3) with |theta3| < 2^(-w).
         Finally for expt+2 < 0 we have:
         t = |x|^(1/k) * (1 + 2*theta3) with |theta3| < 2^(-w).
      */
      err = (expt + 2 > 0) ? expt + 3
        : (expt + 2 == 0) ? 2 : 1;
      /* now t = |x|^(1/k) * (1 + 2^(err-w)) thus the error is at most
         2^(EXP(t) - w + err) */
      if (MPFR_LIKELY (MPFR_CAN_ROUND(t, w - err, MPFR_PREC(y), rnd_mode)))
        break;

      /* If we fail to round correctly, check for an exact result or a
         midpoint result with MPFR_RNDN (regarded as hard-to-round in
         all precisions in order to determine the ternary value). */
      {
        mpfr_t z, zk;

        mpfr_init2 (z, MPFR_PREC(y) + (rnd_mode == MPFR_RNDN));
        mpfr_init2 (zk, MPFR_PREC(x));
        mpfr_set (z, t, MPFR_RNDN);
        inexact = mpfr_pow_ui (zk, z, k, MPFR_RNDN);
        exact_root = !inexact && mpfr_equal_p (zk, absx);
        if (exact_root) /* z is the exact root, thus round z directly */
          inexact = mpfr_set4 (y, z, rnd_mode, MPFR_SIGN (x));
        mpfr_clear (zk);
        mpfr_clear (z);
        if (exact_root)
          break;
      }

      MPFR_ZIV_NEXT (loop, w);
      MPFR_GROUP_REPREC_1(group, w, t);
    }
  MPFR_ZIV_FREE (loop);

  if (!exact_root)
    inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (x));

  MPFR_GROUP_CLEAR(group);
  MPFR_TMP_FREE(marker);
  MPFR_SAVE_EXPO_FREE (expo);

  return mpfr_check_range (y, inexact, rnd_mode);
}

int
mpfr_rootn_si (mpfr_ptr y, mpfr_srcptr x, long k, mpfr_rnd_t rnd_mode)
{
  int inexact;
  MPFR_ZIV_DECL(loop);
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("x[%Pd]=%.*Rg k=%lu rnd=%d",
      mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
     ("y[%Pd]=%.*Rg inexact=%d",
      mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  if (k >= 0)
    return mpfr_rootn_ui (y, x, k, rnd_mode);

  /* Singular values for k < 0 */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
          MPFR_RET_NAN;
        }

      if (MPFR_IS_INF (x)) /* (+Inf)^(1/k) = +0
                              (-Inf)^(1/k) = -0 if k odd
                              (-Inf)^(1/k) = NaN if k even */
        {
          /* Cast k to an unsigned type so that this is well-defined. */
          if (MPFR_IS_NEG (x) && ((unsigned long) k & 1) == 0)
            {
              MPFR_SET_NAN (y);
              MPFR_RET_NAN;
            }
          MPFR_SET_ZERO (y);
          MPFR_SET_SAME_SIGN (y, x);
        }
      else /* x is necessarily 0: (+0)^(1/k) = +Inf
                                  (-0)^(1/k) = +Inf if k even
                                  (-0)^(1/k) = -Inf if k odd */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          MPFR_SET_INF (y);
          /* Cast k to an unsigned type so that this is well-defined. */
          if (MPFR_IS_POS (x) || ((unsigned long) k & 1) == 0)
            MPFR_SET_POS (y);
          else
            MPFR_SET_NEG (y);
          MPFR_SET_DIVBY0 ();
        }
      MPFR_RET (0);
    }

  /* Returns NAN for x < 0 and k even */
  /* Cast k to an unsigned type so that this is well-defined. */
  if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && ((unsigned long) k & 1) == 0))
    {
      MPFR_SET_NAN (y);
      MPFR_RET_NAN;
    }

  /* Special case |x| = 1. Note that if x = -1, then k is odd
     (NaN results have already been filtered), so that y = -1. */
  if (mpfr_cmpabs (x, __gmpfr_one) == 0)
    return mpfr_set (y, x, rnd_mode);

  /* The case k = -1 is probably rare in practice (the user would directly
     do a division if k is a constant, and even mpfr_pow_si is more natural).
     But let's take it into account here, so that in the general case below,
     overflows and underflows will be impossible, and we won't need to test
     and handle the corresponding flags. And let's take the opportunity to
     handle k = -2 as well since mpfr_rec_sqrt is faster than the generic
     mpfr_rootn_si (this is visible when running the trec_sqrt tests with
     mpfr_rootn_si + generic code for k = -2 instead of mpfr_rec_sqrt). */
  /* TODO: If MPFR_WANT_ASSERT >= 2, define a new mpfr_rootn_si function
     so that for k = -2, compute the result with both mpfr_rec_sqrt and
     the generic code, and compare (ditto for mpfr_rec_sqrt), like what
     is done in add1sp.c (mpfr_add1sp and mpfr_add1 results compared). */
  if (k >= -2)
    {
      if (k == -1)
        return mpfr_ui_div (y, 1, x, rnd_mode);
      else
        return mpfr_rec_sqrt (y, x, rnd_mode);
    }

  /* TODO: Should we expand mpfr_root_aux to negative values of k
     and call it if k < -100, a bit like in mpfr_rootn_ui? */

  /* General case */
  {
    mpfr_t t;
    mpfr_prec_t Ny;  /* target precision */
    mpfr_prec_t Nt;  /* working precision */

    /* initial working precision */
    Ny = MPFR_PREC (y);
    Nt = Ny + 10;

    MPFR_SAVE_EXPO_MARK (expo);

    mpfr_init2 (t, Nt);

    MPFR_ZIV_INIT (loop, Nt);
    for (;;)
      {
        /* Compute the root before the division, in particular to avoid
           overflows and underflows.
           Moreover, midpoints are impossible. And an exact case implies
           that |x| is a power of 2; such a case is not the most common
           one, so that we detect it only after MPFR_CAN_ROUND. */

        /* Let's use MPFR_RNDF to avoid the potentially costly detection
           of exact cases in mpfr_rootn_ui (we just lose one bit in the
           final approximation). */
        mpfr_rootn_ui (t, x, - (unsigned long) k, MPFR_RNDF);
        inexact = mpfr_ui_div (t, 1, t, rnd_mode);

        /* The final error is bounded by 5 ulp (see algorithms.tex,
           "Generic error of inverse"), which is <= 2^3 ulp. */
        MPFR_ASSERTD (! MPFR_IS_SINGULAR (t));
        if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - 3, Ny, rnd_mode) ||
                         (inexact == 0 && mpfr_powerof2_raw (x))))
          break;

        MPFR_ZIV_NEXT (loop, Nt);
        mpfr_set_prec (t, Nt);
      }
    MPFR_ZIV_FREE (loop);

    inexact = mpfr_set (y, t, rnd_mode);
    mpfr_clear (t);

    MPFR_SAVE_EXPO_FREE (expo);
    return mpfr_check_range (y, inexact, rnd_mode);
  }
}

int
mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
{
  MPFR_LOG_FUNC
    (("x[%Pd]=%.*Rg k=%lu rnd=%d",
      mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
     ("y[%Pd]=%.*Rg",
      mpfr_get_prec (y), mpfr_log_prec, y));

  /* Like mpfr_rootn_ui... */
  if (MPFR_UNLIKELY (k <= 1))
    {
      if (k == 0)
        {
          /* rootn(x,0) is NaN (IEEE 754-2008). */
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else /* y = x^(1/1) = x */
        return mpfr_set (y, x, rnd_mode);
    }

  if (MPFR_UNLIKELY (MPFR_IS_ZERO (x)))
    {
      /* The only case that may differ from mpfr_rootn_ui. */
      MPFR_SET_ZERO (y);
      MPFR_SET_SAME_SIGN (y, x);
      MPFR_RET (0);
    }
  else
    return mpfr_rootn_ui (y, x, k, rnd_mode);
}